Finite Group Databasebeta

← #36352790

Subgroups — #36352790

Order 22144.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012206C211yesmaximal, characteristic
2#26011843C211yesmaximal, characteristic, complemented
3#26011693C211yesmaximal, characteristic, complemented
4#26011239C211yesmaximal, characteristic, complemented
5D4⋊#13063436C211yesmaximal, characteristic, complemented
6#26011218C211yesmaximal, characteristic, complemented
7#474×C173C211yesmaximal, characteristic, complemented
8C2×#14583813—21no—
9#19149679C2211yescharacteristic
10#19149665—22no—
11#19149653—21no—
12#19149650—21no—
13C4×D692C2211yescharacteristic
14#19149638—21no—
15#19149637—21no—
16C4⋊#13063436C2211yescharacteristic
17#19149603—21no—
18#19149602—22no—
19C4×#13063436C2211yescharacteristic
20C4⋊Q8×C173C2211yescharacteristic
21C4×D4×C173C2211yescharacteristic
22#107×C173—21no—
23C4⋊C8×C173C2211yescharacteristic
24#102×C173—21no—
25C2×Dic346D412yes—
26C2×D692D411yescharacteristic
27C2×C22⋊D173—22no—
28C2×C4×D173—21no—
29#14583813—42no—
30C2×#13063436—21no—
31C2×#13063436—21no—
32C4⋊Dic173—21no—
33C4⋊Dic173—21no—
34C4⋊Dic173D411yescharacteristic
35C22⋊Dic173—22no—
36#14583803—22no—
37#14583803—21no—
38C4×Dic173—22no—
39C2×Q8×C173—21no—
40C2×D4×C173D411yescharacteristic
41C22×C4×C173—21no—
42C2×C8×C173—21no—
43C4⋊C4×C173—21no—
44C4⋊C4×C173—21no—
45C4⋊C4×C173D411yescharacteristic
46C22⋊C4×C173—21no—
47C42×C173C2311yescharacteristic
48C22×D173—21no—
49Dic346—42no—
50D692—21no—
51D692—21no—
52C22⋊D173—82no—
53C2×Dic173—21no—
54C2×Dic173—22no—
55C4×D173—41no—
56C4×D173—41no—
57#13063436—41no—
58#13063436—41no—
59C23×C173—21no—
60Q8×C173—41no—
61Q8×C173—41no—
62D4×C173—21no—
63D4×C173—21no—
64C2×C4×C173—21no—
65C2×C4×C173—41no—
66C2×C4×C173—41no—
67C2×C4×C173—41no—
68C2×C4×C173C2×D411yescharacteristic
69C2×C4×C173C2×D411yescharacteristic
70C2×C4×C173C2×D411yescharacteristic
71C8×C173—41no—
72C22×C173—41no—
73C22×C173—41no—
74C22×C173—41no—
75C22×C173#12011yescharacteristic
76C2×D173—41no—
77C2×D173—41no—
78C2×D173—41no—
79C4×C173—21no—
80C4×C173—41no—
81C4×C173—81no—
82C4×C173#13511yescharacteristic
83C4×C173#13511yescharacteristic
84C4×C173AGL(1,ℤ/8)11yescharacteristic
85C4×C173#13711yescharacteristic
86Dic173—41no—
87Dic173—42no—
88C2×C173—81no—
89C2×C173#44911yescharacteristic
90C2×C173#45211yescharacteristic
91C2×C173#45511yescharacteristic
92D173—81no—
93C173#164611yescharacteristic, complemented, Sylow
94#1646—1731nomaximal, Sylow
95#555—1731no—
96#487—1731no—
97#474D17311yescharacteristic, complemented
98#461—1731no—
99#333—1731no—
100D4⋊C8—1731no—
101D4⋊C8—1731no—
102C2×SD16—3461no—
103C4⋊Q8—1731no—
104C4⋊Q8C2×D17311yescharacteristic
105#124—3462no—
106#121—3461no—
107#121—3461no—
108C4×D4—1731no—
109C4×D4C2×D17311yescharacteristic
110#117—3461no—
111#117—3461no—
112#107—21no—
113C4⋊C8—1731no—
114C4⋊C8C2×D17311yescharacteristic
115#103—3462no—
116#103—3461no—
117#102—21no—
118C4×C8—1731no—
119C2×Q8—21no—
120C2×Q8—1732no—
121C2×D4—3462no—
122C2×D4—1731no—
123C2×D4C22⋊D17311yescharacteristic
124C22×C4—21no—
125C22×C4—3461no—
126SD16—6922no—
127C2×C8—21no—
128C2×C8—3461no—
129C2×C8—3461no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—3461no—
133C4⋊C4—3461no—
134C4⋊C4—1731no—
135C4⋊C4C22⋊D17311yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—3462no—
138C22⋊C4—3462no—
139C22⋊C4—3461no—
140C42—3462no—
141C42C22×D17311yescharacteristic
142C23—21no—
143C23—3461no—
144Q8—41no—
145Q8—41no—
146Q8—6922no—
147D4—21no—
148D4—21no—
149D4—3461no—
150D4—3461no—
151D4—13842no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—3461no—
157C2×C4—3462no—
158C2×C4—6921no—
159C2×C4—6921no—
160C2×C4D4×D17311yescharacteristic
161C2×C4D4×D17311yescharacteristic
162C2×C4C2×C22⋊D17311yescharacteristic
163C8—41no—
164C8—6921no—
165C8—6921no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—6921no—
170C22—6921no—
171C22—6921no—
172C22#1914964711yescharacteristic
173C4—21no—
174C4—41no—
175C4—81no—
176C4—6921no—
177C4—6922no—
178C4#1914969811yescharacteristic
179C4#1914969911yescharacteristic
180C4#1914970311yescharacteristic
181C4#1914971011yescharacteristic
182C2—81no—
183C2—13841no—
184C2#2601163311yescharacteristic
185C2#2601165111yescharacteristic
186C2#2601166711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#474×C173C2
socleC22×C173#120
prime radical#26012206C2
hypercenter#474D173
hypocenterC173#1646
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #26011843C2semidirect
3. #26011693C2semidirect
4. #26011239C2semidirect
5. D4⋊#13063436C2semidirect
6. #26011218C2semidirect
7. #474×C173C2semidirect
93. C173#1646semidirect
97. #474D173semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#260122062
#260118432
#260116932
#260112392
D4⋊#130634362
#260112182
#474×C1732
#1646173